Group Actions, p Groups and Sylow Subgroups

p Groups and Sylow Subgroups

For p prime, G is a p group if the order of every element in G is a power of p.  Note that infinite groups can be p groups.  The infinite direct product of Zp is an example.

If |G| is a power of p then every subgroup, including the powers of x, has order dividing |G|, hence G is a p group.  Conversely, if some other prime q divides |G|, then G contains a q cycle, and is not a p group.  For finite groups, p group means |G| is a power of p.

A sylow subgroup is a maximal subgroup that is also a p group.  Use zorn's lemma to show such a group exists when the containing group is infinite.

Let the p group G have a normal subgroup H.  Since x*H/x is H, G acts on the set H by conjugation.  This action leaves the center of G fixed.  When restricted to H, it fixes H intersect the center of G.  By the fixed point principle, this fixed set has size n, where n = |H| mod p.  Of course H is a p group, so n = 0.  The center of G, intersected with H, is always divisible by p.

As a special case, set H = G, and p divides the number of elements in the center of G.  Therefore the center of every p group is nontrivial.